For a number $l\geq 2$, let ${ \cal{ G}}_l$ denote the family of graphs which have girth $2l+1$ and have no odd hole with length greater than $2l+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{ l\geq 2} { \cal{ G}}_{ l}$ is $3$-colorable. Chudnovsky et al., Wu et al., and Chen showed that every graph in ${ \cal{ G}}_2$, ${ \cal{ G}}_3$ and $\bigcup_{ l\geq 5} { \cal{ G}}_{ l}$ is $3$-colorable respectively. In this paper, we prove that every graph in ${ \cal{ G}}_4$ is $3$-colorable. This confirms Wu, Xu and Xu's conjecture.
Categories: math.CO